A comment on another question reminded me of this old post of Terence Tao's about toy models. I really like the idea of using toy models of a difficult object to understand it better, but I don't know of too many examples since I don't see too many people talk about them. What examples are common in your field, or what examples do you personally think are very revealing? Here's what I've got so far, starting with Terence Tao's example. Feel free to modify any of these examples if I'm not stating them correctly and to elaborate on them in answers if you want.

Fp[t] is a toy model for Z.

Fp[[t]] is a toy model for Zp.

Simplicial complexes are a toy model for topological spaces.

Z/nZ is a toy model for Z (for the purposes of additive number theory).

The DFT is a toy model for the Fourier transform on the circle.

Which properties of the original objects carry over to your toy model, and which don't? As usual, stick to one example per post.

Are simplicial complexes really toy models for topological spaces? While the category of simplicial complexes has numerous problems, they are mostly confined to the morphisms -- most spaces of interest are at least homotopy equivalent to simplicial complexes.
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Andy PutmanOct 20 '09 at 3:37

Well, I wouldn't really know; feel free to edit that one or write an answer.
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Qiaochu YuanOct 20 '09 at 3:42

Not sure if this is the right place to ask, but when you say F<sub>p</sub>[t] is a toy model for Z I guess you are referring to the general philosophy that problems over function fields are easier to deal with than those over number fields. Can someone actually elaborate on this analogy between number fields and function fields? I'm not sure where I can find information about this. Ring of integers being Dedekind, finite residue field, RH over function fields easier to deal with, anything else?
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Ho Chung SiuOct 20 '09 at 4:07

I have the same question; I think it should probably be asked separately.
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Qiaochu YuanOct 20 '09 at 4:14

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I think the category of CW complexes (c.f. Hatcher's book) is a toy model for topological spaces.
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john mangualDec 27 '09 at 15:38

15 Answers
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One can think of toric varieties as "toy examples" of algebraic varieties. A lot can be said about them via combinatorial data, but they definitely are special (they are always rational varieties, for example).

The best examples I've come up with come from rational homotopy theory--commutative differential graded Q-algebras as a toy model for spaces and chain complexes of Q-vector spaces as a toy model for spectra--though really this is an instance of "toy examples" because we can build actual spaces/spectra corresponding to these algebraic data. It feels a bit like a toy model, though, I guess because those spaces aren't very geometric.

Perhaps this doesn't count as a toy model, rather a toy example. A nice basic example for GIT is n points in CP1 under the action of SL(2,C). A lot of the usual elements of the theory look nice in this picture. For example, the Hilbert Mumford criterion shows that a collection of n points is semi-stable iff all points have multiplicity less or equal to n/2 whilst a collection of n points is stable iff all points have multiplicity strictly less than n/2.

It's also a nice example of the equivalence between symplectic reduction and the GIT quotient. If you fix a Fubini-Study metric on CP1 and look at the action of the corresponding SU(2) you can ask for a moment map. Thinking of CP1 as a coadjoint orbit in su(2)*, the moment map takes n points to their centre of mass. Now the equivalence of symplectic and GIT quotients says that, provided we don't have two points each of multiplicity n/2, you can move n points in CP1 by an element of SL(2,C) so that their centre of mass is zero if and only if all multiplicities are strictly less than n/2. (The case when two points each have multiplicity n/2 is special because of the additional C* stabiliser.) In one direction this is obvious, but in the other I think this is a neat non-trivial statement (at least when n is large).

Your first example of Fp[t] being a toy model for Z is a facet of the well known (to number theorists) analogy between Q and Fp(t). This analogy extends fabulously to a more general connection between the arithmetic of number fields, i.e. finite extensions of Q, and of function fields, i.e. finite extensions of Fp(t).

Derived categories of finite dimensional hereditary algebras for other derived categories. (Though under good circumstances other derived categories can turn out to be equivalent to derived categories of f.d. hereditary algebras.)

This is not so much a new example of toy model but rather a complement to the excellent post by Tao on dyadic models. Like most good ideas, it has been rediscovered many times, in many different areas, and given different names. In quantum field theory and in particular in the framework of Wilson's renormalization group, many models have a Euclidean version (on $\mathbb{R}^d$) and a hierarchical version (for instance on $\mathbb{Q}_p^d$). The so-called hierarchical model in physics is another instance of what Tao calls "dyadic models". Wilson himself referred to it as "the approximate RG recursion" and it played a key role in his path to discovery of his RG theory (see quote by Wilson on page 8 of this paper). A good reference on the hierarchical model in physics is this review by Meurice. In probability theory, the dyadic model is the branching random walk or Brownian motion/Mandelbrot cascades used as a toy model for the Gaussian free field, see the references in this MO post.

But a "real computer," no matter how many networked machines it's made of, how many processors each machine has, or what kind of file system it's running, is still just a finite state automaton, so you could just as well say that "real computers" are examples from a class of toy models for Turing machines.
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VectornautDec 15 '14 at 21:34